Grothendieck groups of complexes with null-homotopies

TL;DR

This paper provides a simplified proof of a theorem relating relative K-groups of chain complexes with bounded length to those with an absolute length bound, clarifying their equivalence.

Contribution

It offers a streamlined proof of Foxby and Halvorsen's theorem connecting different formulations of K-groups for chain complexes.

Findings

Relative and absolute bounded chain complex K-groups are equivalent

Simplified proof enhances understanding of the theorem

Clarifies the relationship between different length bounds in chain complexes

Abstract

We give a streamlined proof of a theorem of Foxby and Halvorsen. The theorem states that certain relative K-groups made from chain complexes with bounded (but arbitrarily long) length coincide with similar K-groups in which one sets an absolute bound on the length of the complexes.